Wave Turbulence in Shallow Water Models

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The comparison between different shallow water models, with different degree of shallowness (and of dispersion) is therefore of interest, e.g., for the study of waves in basins with inhomogeneous depth. In the shallow water regime there are several models that can be considered to describe the ocean surface. There is the linear theory (see, e.g., [38]) which can pre- dict the dispersion relation of small amplitude waves, but which is insufficient to study turbulence. There are also non-linear theories, such as the shallow water model [39] FIG. 1. The shallow water geometry considered in the sim- for non-dispersive waves, as well as the Boussinesq model ulations: x and y are the horizontal coordinates, and z is [40] for weakly dispersive waves which is the one used in the vertical coordinate. The surface height is h, with h0 the some of the most recent works on the subject [34]. While height of the fluid column at rest. The fluid surface is at pressure p0. L is a characteristic horizontal length (assumed to the former non-linear model is valid in the strict shal- be much larger than h0). Gravity acts on the −zˆ direction low water limit, the latter can be used in cases in which and its acceleration has a value of g. the wavelengths are closer to (albeit still larger than) the depth of the basin. In the present work, we study turbulent solutions of branch corresponding to non-dispersive waves, and an- the shallow water model and of the Boussinesq equations other corresponding to dispersive waves. We interpret using direct numerical simulations. Previous numerical this as the result of short wavelength waves seeing an ef- studies considered the Hamiltonian equations for a po- fectively deeper flow resulting from the interaction with tential flow with a truncated non-linear term, or the ki- waves with very long wavelength. (e) The probability netic equations resulting from weak turbulence theory at density function of surface height can be approximated moderate spatial resolution [32, 41] (with the notable ex- by both skewed normal and Tayfun distributions. In the ception of [42]). Here, we solve the primitive equations, latter case, the parameters of the distribution are com- without truncating the non-linear terms, potentially al- patible with those previously found in observations and lowing for the development of vortical motions and of experiments [37]. strong interactions between waves, and with spatial resolutions up to 20482 grid points. The paper is organized as follows. In section II we introduce both models, show the assumptions made in II. THE SHALLOW WATER AND BOUSSINESQ order to obtain them, derive their energy balance equa- EQUATIONS tions, and briefly discuss the predictions obtained in the framework of weak turbulence theory. In section III we Let us consider a volume of an incompressible fluid describe the numerical methods employed and the simu- with uniform and constant (in time) density, with its lations. Then, in section IV we introduce several dimen- bottom surface in contact with a flat and rigid bound- sionless numbers defined to characterize the flows, and ary, and free surface at pressure p0. A sketch illustrating present the numerical analysis and results. We present the configuration is shown in Fig. 1; x and y are the hori- wavenumber energy spectra and fluxes, time resolved zontal coordinates, z is the vertical one, h is the height of spectra (as a function of the wavenumber and frequency), the fluid column (i.e., the z value at the free surface), h0 frequency spectra, and probability density functions of is the height of the fluid column at rest, L is a character- the fluid free surface height. Finally, in section V we istic horizontal length, gravity acts on the zˆ direction − present the conclusions. The most important results are: and its value is given by g. It is assumed that L h0 as (a) As in previous experimental and observational studies we are interested in shallow flows. ≫ [35, 36] we find now in simulations that different regimes In the inviscid case, both the Euler equation and the arise depending on the fluid depth and the degree of non- incompressibility condition hold in the fluid body, linearity of the system. (b) We obtain a power spectrum of the surface height compatible (within statistical uncer- ∂v 1 tainties) with k−2 in the shallow water (non-dispersive) + (v ∇)v = ∇p gz,ˆ (1) case, and one∼ compatible with a k−4/3 spectrum as the ∂t · −ρ − fluid depth is increased using the∼ Boussinesq (weakly dis- ∇ v =0. (2) · persive) model. The latter spectrum is also compatible with predictions of weak turbulence theory [34]. (c) Dis- Under certain assumptions, to be discussed in the follow- persion in the Boussinesq model results in more promi- ing paragraphs, the evolution of the free surface can be nent small scale features and the development of rapidly adequately described by means of a vector equation for varying waves. (d) In the weakly dispersive regime, the the two horizontal components of the velocity at the in- non-linear dispersion relation obtained from the simula- terface, plus an equation for the local height of the fluid tions has two branches in a range of wavenumbers, one column. 3

A. Linear dispersion relation If we linearize these equations, we ﬁnd again the dispersion relation given by Eq. (4), as expected. In the Considering the case of very small amplitude waves, presence of external forcing F, and viscosity ν, the equa- one can linearize the system of Eqs. (1) and (2) (see, tions can be written as e.g., [38]). The solutions of the resulting equations are ∂u ν gravity waves with the following dispersion relation = (u ∇)u g∇h + ∇ (h∇u)+ F, (9) ∂t − · − h ·

−2kh0 ∂h 2 1 e = ∇ (hu). (10) ω = gk − − . (3) ∂t − · 1+ e 2kh0 We will refer to this set of equations as the shallow water We are interested in the shallow water case, i.e., when model, or SW model. In these equations the viscosity ν h L h k 1. In that limit the following disper- 0 ≪ ⇒ 0 ≪ was added to the horizontal velocity ﬁeld u, which be- sion relation results haves as a compressible ﬂow (i.e., it has non-zero divergence, see [43]). This choice of the viscous term ensures ω = gh k = c k, (4) 0 0 conservation of the momentum hu, and also that energy p dissipation is always negative, as in Sec. II D. where c0 = √gh0 is the phase velocity. Note in this case waves are not dispersive, unlike the general case given by Eq. (3). C. Boussinesq model

B. Shallow water model As the depth of the fluid increases, dispersion becomes important. There are several models that introduce weak dispersive effects perturbatively, but many are built to It is possible to derive a set of non-linear equations study waves propagating in only one direction. The for the surface height and the horizontal velocity at the Boussinesq equations for surface waves (see, e.g., [40, 44]) surface by using the fact that the fluid layer is shallow. provide a model to study weakly dispersive waves prop- Considering the characteristic magnitudes of all quanti- agating in any direction. This model not only broadens ties in Eq. (1) (L, p , h , g, etc.), and using the fact that 0 0 the range of physical phenomena encompassed by the SW in a shallow flow h k 1 with k = 2π/L, one obtains 0 model, but adding dispersive effects also makes it more hydrostatic balance in≪ the vertical direction (for further enticing to weak turbulence theory, for which dispersion details, see [39]), which results in the pressure profile effects are of the utmost importance. Let us take a look at the first terms in the Taylor ex- p = ρg(h z)+ p0. (5) − pansion of the dispersion relation in Eq. (3), As h is not a function of z, neither will be the horizontal 2 2 2 1 2 2 4 pressure gradient and the horizontal components of the ω = c0k c0h0k + ..., (11) velocity (as long as they do not depend initially on z). − 3 In this way, the horizontal components of Eq. (1) can be where the first term is the non-dispersive shallow water written as term. The idea is to add terms to Eqs. (9) and (10) such that the linear dispersion relation of the new sys- ∂u = (u ∇)u g∇h. (6) tem coincides, up to the fourth order, with the expan- ∂t − · − sion in Eq. (11). This can be done by adding the term h2 2∂ u/3 to Eq. (9), resulting in the following system, where u(x,y,t) = vxxˆ + vyyˆ is the horizontal velocity, 0∇ t and ∇ is now the horizontal gradient. ∂u 1 2 2 ∂u Using the fact that vx and vy are independent of z we = (u ∇)u g∇h + h ∂t − · − 3 0∇ ∂t can integrate the incompressibility condition, obtaining ν + ∇ (h∇u)+ F, (12) h · ∂vx ∂vy vz(x,y,z,t)= z + +˜vz(x,y,t). (7) ∂h − ∂x ∂y = ∇ (hu). (13) ∂t − · Finally, by taking the appropriate boundary conditions We will refer to this system as the Boussinesq model, or and setting z = h(x,y,t), Eq. (7) provides an equation BQ model. For F =0 and ν =0, the dispersion relation for the evolution of the height of the fluid column, namely obtained by linearizing these equations is ∂h ∂ ∂ c k + (hvx)+ (hvy)=0. (8) ω = 0 , (14) ∂t ∂x ∂y 2 2 h0k 1+ 3 Note that we do not have to assume irrotationality to q derive neither Eq. (6) nor Eq. (8). which, up to the fourth order, coincides with Eq. (3). 4

Note that there are other choices for the extra term in as the mean potential energy, such that the sum of both Eq. (9) that result in many formulations of the Boussi- gives the mean total energy E. nesq model, all compatible up to fourth order in a Tay- The dispersive term present in the BQ model changes lor expansion in terms of h0k [40]. The formulation we the balance given by Eq. (17). However, since the extra 2 use here was employed in previous studies of wave tur- term is of order (h0/L) , as long as we are in a sufficiently bulence [34], and is also easy to solve numerically using shallow flow it will be very small, and therefore, negligible pseudospectral methods by writing Eq. (12) as for the conservation of energy. We verified this is the case in our numerical simulations. ∂u′ ν = (u ∇)u g∇h + ∇ (h∇u)+ F, (15) ∂t − · − h · ′ 2 2 E. Weak Turbulence prediction where u = u, and where = (1 h0 /3) is the Helmholtz operator.H This operatorH − can be∇ easily in- verted in Fourier space [45, 46], and the resulting equa- We briefly present some results obtained in the frame- tions can be efficiently solved by means of pseudospectral work of weak turbulence theory for the BQ model (as codes. It is interesting that the same operator appears in the derivation is a bit cumbersome, only a general out- Lagrangian-averaged models [47]. In these models, and in line will be given here; please see [34] for details). Weak regularized versions of the shallow water equations [48], turbulence is studied in the BQ model assuming the fluid it introduces dispersion that results in an accumulation is inviscid and irrotational, so that the velocity can be of energy at small scales [49]. written in terms of a velocity potential. To obtain a statistical description of the wave field, it is also assumed that it is homogeneous and that the free modes are un- D. Energy balance correlated. At first sight, the quadratic nonlinear terms in An exact energy balance can be easily derived for the Eqs. (12) and (13) indicate modes interact in triads, with SW model. The equation is useful to verify conservation the wave vectors of the three interacting modes lying over in pseudospectral codes. By taking the dot product of a triangle, and the three frequencies satisfying the reso- Eq. (9) and hu, setting F = 0, and using Eq. (10), we nant condition (see, e.g., [12]) obtain k = p + q (22) ∂ hu2 h2 hu2 ω(k)= ω(p)+ ω(q). (23) + g = ∇ u + gh2u ∂t 2 2 − · 2 (16) However, as there are no three wave vectors k, p, q that + νu [∇ (h∇u)]. · · satisfy these two conditions when the dispersion relation Integrating in x and y over an area A and taking periodic is given by Eq. (11), three wave interactions are forbid- boundary conditions yields den. Thus, only four wave interactions are present (which do satisfy their corresponding condition). dE After a transformation of the fields, it is possible to = 2νZ, (17) dt − write an equation for the evolution of the two-point correlator of the transformed fields. This is the so-called where kinetic equation, and has the following form 1 hu2 h2 E = + g dxdy (18) ∂N0 2 A ZZ 2 2 =4π T0,1,2,3 N0N1N2N3 ∂t Z | | is the mean total energy, and 1 1 1 1 + (24) 1 h ∇u 2 N0 N1 − N2 − N3 Z = | | dxdy (19) δ(k + k k k ) A ZZ 2 0 0 − 2 − 3 δ(ω + ω ω ω )dk , is a mean pseudo-enstrophy, such that 2νZ is the mean 0 0 − 2 − 3 123 − energy dissipation rate. As h is always positive, the en- where Ni = N(ki) is the wave action spectral den- ergy dissipation is always negative. The total energy is sity (i.e., the two-point correlator of the wave action, conserved when ν =0. the latter being a quantity proportional to the surface Now we can define height), the deltas express the fact that interactions are 1 hu2 between four wave vectors and their associated frequen- U = dxdy (20) cies, T is the coupling coefficient between the four A ZZ 2 0,1,2,3 modes, and dk123 = dk1dk2dk3. From this equation, di- as the mean kinetic energy, and mensional analysis yields the following expression for the energy spectrum 1 h2 − V = g dxdy (21) E(k) k 4/3. (25) A ZZ 2 ∼ 5

III. NUMERICAL SIMULATIONS

0.142135 We performed several numerical simulations of both the shallow water and the Boussinesq models. These were done using the GHOST code [51–53], which uses a pseudospectral method with periodic boundary condi- ×10−7 0 142130 2 0 tions on a L L = 2π 2π sized box (with L the E . . 0 0 0 ǫ 1.5 × ×

+ box length), the “2/3 rule” for the dealiasing [54], ex- 1.0 0 5 plicit second order Runge-Kutta for time stepping, and νZ . 2 0.0 is parallelized using MPI and OpenMP. Almost all sim- − −0.5 2 2 0.142125 −1 0 ulations shown here were done on grids of N = 2058 dt . dE −1 5 2 2 2 . 0 50 100 150 200 points, with a few on grids of N = 1024 or 512 points t (with N the linear resolution). As a result of dealiasing, 0 50 100 150 200 the maximum resolved wavenumber is t kmax = N/3. (26) FIG. 2. (Color online) Total energy as a function of time Note all magnitudes in the code are dimensionless, with for simulation A06 (see Table I). As the fluid starts from rest the smallest wavenumber kmin = 2π/L0 = 1, and the and the forcing is applied, energy increases until it reaches a largest wavenumber kmax = 2π/λmin being associated turbulent steady state (note energy at t = 0 is different from with the minimum resolved scale λmin. zero, as the flow potential energy is never zero). All the analysis of the simulations was performed after the simulations All runs are direct numerical simulations, with all rele- reached the turbulent steady state. Inset: Energy balance as vant space and time scales resolved explicitly. The pseu- a function of time (see Eq. (28)). Note the balance is satisfied dospectral method with the 2/3 rule is equivalent to a up to the seventh decimal place. purely spectral method [54]: it converges exponentially fast, it conserves all quadratic invariants of the equations (i.e., there is no numerical dissipation introduced by the method), and it also has no numerical dispersion. All From this spectrum and using dimensional analysis, it is this was verified explicitly during the development of the easy to show that in the presence of dissipation, the dis- code, using several test problems for the SW and BQ sipation wavenumber in such a flow is k [ǫ/(h2ν3)]1/5, η ∼ 0 equations. where ǫ is the mean energy injection rate. Most previous numerical studies on wave turbulence − A scaling compatible with a k 4/3 spectrum was in gravity waves were done at lower resolutions, with the ∼ observed in laboratory and field datasets [35, 36], where exception of [42]. But the key difference between pre- − a spectrum compatible with k 2 was also found in vious simulations and the ones presented here (besides ∼ shallower regions of the fluid. the fact that these are for shallow flows, not for deep The prediction in Eq. (25) applies to the BQ model, flows) is that the physical model we use does not assume when dispersion is not negligible. Before proceeding, we potential flow, and, more importantly, we do not trun- should comment on some peculiarities of the SW model cate the non-linear term, thus retaining all high order regarding wave turbulence. First, an inspection of its non-linearities. Another difference is that we do not in- dispersion relation, Eq. (4), indicates that three wave troduce an artificial dissipation term as it is usually done, interactions are possible in this model, and as a result but one based on physical grounds. The key motivation the arguments above for four-wave interactions do not for these choices is to be able to compare with experi- apply. Weak turbulence theory can be used in systems ments in the future, where vortical motions can develop, with three-waves interactions (with the case of deep wa- and where dissipation also plays a non-negligible role. To ter flows being a paradigmatic one, but see also the case achieve higher resolutions than the ones studied here be- of rotating [14] and of magnetohydronamic [15] flows). comes increasingly more expensive as the BQ model is However, the SW model is non-dispersive, and as a re- dispersive. sult the resonance condition is only satisfied for collinear All the simulations were started from the fluid at rest. wave vectors. Resonant interactions can then only cou- An external mechanical forcing injected energy in the ple modes that propagate in the same direction (i.e., system, allowing it to reach for sufficiently long times an along the ray of the wave), and non-resonant interactions out-of-equilibrium turbulent steady state, after an initial must be taken into account to consider other couplings. transient. To excite waves, and prevent external injec- But more importantly, dispersion is crucial in weak tur- tion of energy into vortical motions, the forcing had the bulence theory to have decorrelation between different following form waves: without dispersion, all modes propagate with the F = ∇f, (27) same velocity, and the modes initially correlated remain correlated for all times (see, e.g., [50] for a discussion of where f is a randomly generated scalar function, with a these effects in the context of acoustic turbulence). time correlation of one time unit, amplitude f0, and ap- 6

Simulation Re Fr Ds Nl h0/L0 f0/U0 [kf1 ,kf2 ] N − − A01 260 0.005 0.27 1.2 × 10 5 8.0 × 10 4 0.76 [3,8] 1024 A02 370 0.0059 0.22 1.6 × 10−5 6.4 × 10−4 0.71 [3,8] 1024 − − A03 820 0.0075 0.33 2.6 × 10 5 4.9 × 10 4 0.64 [3,8] 2048 − − A04 760 0.0067 0.36 2.2 × 10 5 5.3 × 10 4 0.69 [3,8] 2048 A05 760 0.007 0.33 2.3 × 10−5 4.8 × 10−4 0.69 [3,8] 2048 − − A06 360 0.0066 0.33 2.1 × 10 5 4.8 × 10 4 0.73 [3,8] 2048 A07 570 0.0091 0.43 4.1 × 10−5 6.4 × 10−4 0.92 [3,8] 2048 − − A08 350 0.0083 0.43 3.4 × 10 5 6.4 × 10 4 1 [3,8] 2048 − − A09 290 0.0086 0.43 3.6 × 10 5 6.4 × 10 4 0.98 [3,8] 2048 A10 420 0.012 0.43 7.8 × 10−5 6.4 × 10−4 1.4 [3,8] 2048

TABLE I. Dimensionless numbers (deﬁned in the text) and parameters for runs in set A. Re is the Reynolds number, Fr is the Froude number, Ds is the dispersivity, Nl is the non-linear number, h0/L0 is the height of the ﬂuid at rest divided by the length of the box, f0/U0 the amplitude of the forcing divided by the rms speed, kf1 and kf2 are respectively the minimum and maximum wavenumbers in which the random forcing is applied, and N is the linear resolution. In all cases, the Boussinesq model was solved.

Simulation Re Fr Ds Nl h0/L0 f0/U0 [kf1 ,kf2 ] N − − B01 5600 0.022 0.14 2.5 × 10 4 8.0 × 10 4 0.45 [1,5] 512 − − B02 3700 0.015 0.14 1.1 × 10 4 8.0 × 10 4 0.34 [1,5] 512 B03 5000 0.012 0.14 7.2 × 10−5 8.0 × 10−4 0.29 [1,5] 512 − − B04 7100 0.013 0.11 9.1 × 10 5 3.2 × 10 4 0.24 [1,5] 1024 − − B05 830 0.005 0.11 1.2 × 10 5 3.2 × 10 4 0.8 [3,8] 1024 − − B06 1200 0.011 0.11 6.2 × 10 5 3.2 × 10 4 0.57 [3,8] 1024 − − B07 120 0.0046 0.14 1.0 × 10 5 8.0 × 10 4 0.82 [3,8] 512 B08 980 0.012 0.17 7.9 × 10−5 2.5 × 10−4 0.54 [3,8] 2048 − − B09 2500 0.038 0.27 7.4 × 10 4 8.0 × 10 4 0.31 [3,8] 1024 − − B10 670 0.0042 0.17 8.5 × 10 6 2.5 × 10 4 0.79 [3,8] 2048 −6 −4 BSW 11 100 0.0039 0.14 7.6 × 10 8.0 × 10 0.96 [3,8] 512 −5 −4 BSW 12 470 0.013 0.11 9.0 × 10 3.2 × 10 0.24 [1,5] 1024

TABLE II. Dimensionless numbers and parameters for runs in set B. Labels are as in Table I. The Boussinesq model was solved in all cases except for runs BSW 11 and BSW 12, that were done solving the shallow water model.

plied in a band of wavenumbers in Fourier space between energy associated with the equilibrium height h0). The kf1 and kf2 (see Tables I, II, and III). Note that having total energy then grows under the action of the external a mechanical forcing in the momentum equation adds an mechanical forcing, and after t 80 the system reaches extra term to the right hand side of Eq. (17), a turbulent steady state in which≈ the energy fluctuates around a mean value, and in which the energy injection dE = 2νZ + ǫ, (28) and dissipation are equilibrated on the average. Even dt − though pseudospectral methods are known to introduce no numerical dissipation, in the inset of Fig. 2 we also where the mean energy injection rate can be computed show explicitly that the energy balance (Eq. (28)) is sat- as − isfied with an error of order 10 7, which remains stable 1 and does not grow even after integrating for very long ǫ = hu fdxdy. (29) A ZZ · times. A Under the procedure described above, the typical evolution of the energy in a numerical simulation is shown in Fig. 2. The energy starts from the value corresponding To ensure that the flow in the simulations remained to the fluid at rest (i.e., all the energy is the potential shallow for all excited wavenumbers, we enforced the fol- 7

Simulation Re Fr Ds Nl h0/L0 f0/U0 [kf1 ,kf2 ] N − − C01 1400 0.0067 0.27 1.9 × 10 5 8.0 × 10 4 0.56 [3,8] 1024 C02 1400 0.0073 0.24 2.4 × 10−5 7.0 × 10−4 0.55 [3,8] 1024 − − C03 1900 0.0092 0.31 4.1 × 10 5 4.6 × 10 4 0.54 [3,8] 2048 − − C04 1400 0.0057 0.43 1.6 × 10 5 6.4 × 10 4 0.74 [3,8] 2048 C05 470 0.0067 0.54 2.2 × 10−5 8.0 × 10−4 1.1 [3,8] 2048

TABLE III. Dimensionless numbers and parameters for runs in set C. Labels are as in Table I. The Boussinesq model was solved in all runs.

lowing condition where Lf is the forcing scale (defined as 2π/kf0 ), and what we will call the dispersivity, Ds, defined as h0 kmax = h0 < 1 λmin 2π 2πh0 Nh0 Ds = h0kmax = = (34) 6π λmin 6π h0 < . (30) ⇒ N following Eq. (30). This last number, only relevant for where λmin is, as already mentioned, the shortest wave- the Boussinesq model, measures how strong the dis- length resolved by the code in virtue of the condition persion is at the smallest scales, and for sufficiently given by Eq. (26). small Ds we can expect the solutions of the Boussinesq model to converge to the solutions of the shallow water model. In fact, it is easy to show from the weak tur- IV. RESULTS bulence spectrum in Eq. (25) that when the maximum resolved wavenumber kmax is associated with the dissi- A. Description and classification of the simulations pation wavenumber kη, then

U0Lf 5/3 The spectral behavior of the flow in the simulations de- Re 1/3 Ds . (35) ∼ h0ǫ pends on the external parameters. We can independently control the height of the fluid at rest h0, the viscosity ν, Decreasing Ds below the value given by this relation the gravity acceleration g, the amplitude of the forcing should result in negligible dispersion at all resolved f0, the range of wavenumbers in which the force is ap- wavenumbers. Note that the level of dispersion in a given plied, and the linear resolution N. However, all these Boussinesq run depends on the wavenumber, and Ds ac- parameters can be reduced to a smaller set of dimension- tually quantifies the strongest possible dispersion at the less controlling parameters. smallest scales in the flow. One of these parameters is the Froude number By qualitatively assessing each run, we can classify them into three sets, A, B, and C. In tables I, II, III U0 the different dimensionless parameters, along with a few Fr = , (31) √gh0 other useful quantities, are given for each simulation in each set, respectively. How and why these three sets dif- which measures the ratio of inertia to gravity acceler- fer from each other will be made clear in the following ation in the momentum equation, and where U0 is the sections, when we discuss the actual results. But, for the r.m.s. velocity. moment, it is fruitful to analyze the behavior of the val- Another dimensionless parameter is the non-linear ues of Re and Ds in each set, so as to keep them in mind number, Nl. In order to be in the regime of weak turbu- for later on. lence, nonlinearities should be small. The effect of non- The values of Re and Ds for all the Boussinessq runs linearities can be measured by how large perturbations are shown in Fig. 3. As a reference, Fig. 3 also shows the 1/3 in h are compared to h0, so we define Nl as curve given by Eq. (35) with U0Lf /(h0ǫ ) estimated from the values from the simulations in set A. Points hrms h0 Nl = − , (32) below that curve are expected to have non-negligible dis- h0 persion. Runs in set A have relatively small Re (. 1000), and D varying between 0.02 and 0.05. In other where hrms is the r.m.s. value of h. s words, dispersion effects in≈ runs in set≈A are important. The two remaining dimensionless numbers are the Reynolds number, Runs in set B have smaller values of Ds (except for one run with D 0.27, all other runs have D < 0.2), and s ≈ s U L Re varying between 100 and 7000. These runs have Re = 0 f , (33) ≈ ≈ ν small or negligible dispersion, and note all the SW runs 8

8000 weak turbulence. On the other hand, the run in set B − A displays an inertial range compatible with k 2 depen- 7000 ∼ B dency. While this spectrum is not predicted by weak 6000 C turbulence, it was observed before in experiments and observations [36]. The run in group C shows a shallower 5000 spectrum with no clear inertial range. We think of runs in this set as transitional between the other two. 4000

Re The other runs in sets A, B, and C show similar power 3000 spectra for h. To show this, we present the compensated spectra for the simulations in sets A and B in 2000 Figs. 6 and 7 respectively (simulations from set C do not have a clearly defined inertial range and are therefore not 1000 shown). The simulations from set A are compensated by h2/3ǫ2/3k−4/3 (which is the weak turbulence spectrum, 0 0 0.1 0.2 0.3 0.4 0.5 using the height of the fluid column at rest, h0, and the Ds energy injection rate, ǫ, as prefactors), while the ones in −2 set B are compensated by gh0k (more details on the FIG. 3. (Color online) Values of the Reynolds number Re choice of the prefactor are given below). These figures and dispersivity Ds for all the Boussinesq runs, separated into indicate that, within statistical uncertainties, all spectra three sets, A (circles in the gray/red region), B (triangles in in each set collapse to the same power laws, and that the the dark/blue region), and C (stars in the light/green region), simulations are well converged from the point of view according to their different spectral behavior as discussed in of spatial resolution. Furthermore, we verified that the Sec. IV B. The boundaries separating the three regions are energy flux is approximately constant in the scales corre- ∼ 5/3 arbitrary. The solid white curve corresponds to Re Ds ; sponding to the inertial range of each simulation. Within points below that curve are expected to have non-negligible the limitations of spatial resolution and the drop in the dispersion (see Eq. (35)). Note that runs in set A have rela- flux for large wavenumbers caused by viscous dissipation, tively small Re but larger dispersion, while runs in set B have an incipient inertial range can be identified in the flux of either small or negligible dispersion. each simulation. Figure 8 shows the instantaneous energy flux (normalized by the energy injection rate ǫ averaged over time) as a function of k for several simulations in we performed belong to this set. The runs in set C are sets A and B. The energy flux Π(k) was calculated from intermediate between these two regimes. the energy balance equation in Fourier space, as is usu- Finally, although the mechanical forcing we use intro- ally done for turbulent flows. Figure 8 also shows the duces no vorticity in the horizontal velocity field, some normalized energy dissipation rate as a function of time vorticity is spontaneously generated as the flow evolves. (equivalent to the normalized energy flux as a function This is probably also the case in experiments. In order of time) for the same runs, to show that this quantity to quantify the presence of vortical structures, we calcu- fluctuates around a mean value in the turbulent steady lated the ratio of vorticity to divergence in the horizontal velocity field state. The kinetic energy spectrum is similar to the power ∇ u spectrum of h, and in approximate equipartition with the h| × |i, (36) ∇ u potential energy spectrum once the system reaches a tur- h| · |i bulent steady state. It is interesting to analyse this in the which turns out to be 0.1 for all simulations. As a light of the values of the dimensionless parameters in the ≈ result, although the flow is not perfectly irrotational, the runs as shown in Fig. 3. As was explained in the previous amplitude of vortical modes is small compared with the section, set A corresponds to runs with lower Reynolds amplitude of modes associated with the waves. number and larger dispersivity (Re . 1000, and Ds varying between 0.02 and 0.05). As a result, these runs can be expected≈ to display≈ weak turbulence behavior as B. Energy spectra described in Section II E, because the nonlinearities are not so large as to break the weak turbulence hypothesis The power spectrum of h (proportional to the spec- [34], and the dispersion is not so low as to render the trum of the potential energy) as a function of the wave higher order terms of Eq. (14) negligible (in which case number is shown in Fig. 4 for runs A06, B08, and C02. four-wave interactions would no longer be dominant, and Figure 5 shows a close-up of the same spectrum in the the hypothesis used to derive Eq. (24) would not be sat- inertial range. It is clearly seen that runs in each set isfied). In contrast, runs in set B have larger Re and show a different behavior. On the one hand, the run lower D (except for one run with D 0.27, all other s s ≈ belonging to group A has an inertial range compatible runs have Ds < 0.2, and Re varying between 100 and with k−4/3 scaling, which is the spectra predicted by 7000). In this case dispersion is smaller or≈ negligible, ∼ ≈ 9

10−9 102 10−10 10−11

−12 3

10 / 4

−13 − 10 1

k 10

−14 3 10 / 2 ) −15 ǫ 3

k 10 / ( 2 0 h 10−16 E −17 A06 /h 10 ) A02

k 100 10−18 B08 ( A06 h 10−19 C02 A08 gE −4 3 10−20 ∼ k / A09 − − 10 21 k 2 A10 − 10−22 10 1 100 101 102 100 101 102 k k

FIG. 4. (Color online) Power spectrum of h (proportional FIG. 6. (Color online) Compensated spectrum of potential to the spectrum of the potential energy) for runs A06 (BQ energy for several simulations in set A. The spectra are com- 2 2 3 model, 2048 grid points, Re = 360, and Ds = 0.33), B08 (BQ / 2/3 −4/3 2 pensated by h0 ǫ k . The average slope for all the runs model, 2048 grid points, Re = 980, and Ds = 0.17), and C02 − ± 2 is 1.34 0.12. (BQ model, 1024 grid points, Re = 1430, and Ds = 0.24). ∼ −4/3 ∼ −2 Two power laws, k and k , are shown as references. 10−5

10−6

−10

10 2 −

k −7

0 10

−11

10 /gh

) B04 ) k −8

( 10 k B06 h ( h −12 B07 E 10 A06 gE −9 B08 B08 10 BSW 11 C02 10−13 BSW 12 −4 3 ∼ k / 10−10 100 101 102 −2 k k 10−14 100 101 102 k FIG. 7. (Color online) Compensated spectrum of potential energy for several simulations in set B. The spectra are com- −2 FIG. 5. (Color online) Detail of the three spectra in Fig. 4 pensated by gh0k . The average slope for all the runs is for a subset of wavenumbers to show the inertial range of the −2.18 ± 0.29. runs. Note the scaling of runs A06 and B08.

fronts in the velocity. Such a field would actually have a spectrum k−2 (note this is also the behavior expected while nonlinearities can be expected to be larger, two ∼ conditions that render the derivation resulting in Eq. (25) for two-dimensional non-dispersive acoustic turbulence invalid. [50], that also develops sharp fronts). The spectrum can be obtained from dimensional analysis and the scal- Moreover, the gh k−2 spectra observed in the sim- ∼ 0 ing that results for the energy is equivalent to Phillips’ ulations in set B include those runs that solve the SW spectrum [27] but in two dimensions. In the presence of model. Therefore, these spectra cannot be explained by strong nonlinearities, we can assume that the nonlinear weak turbulence, as SW simulations have no dispersion and gravity terms are of the same order, and the arguments in Section II E do not apply. Also, note that in the non-dispersive limit, for constant and u ∇u g∇h. (37) fixed h, the SW equations can be reduced to the two- · ∼ dimensional Burgers equations, which amplify negative It is also reasonable to assume that the kinetic and po- field gradients by strong nonlinearities resulting in sharp tential energies will be of the same order (i.e., in equipar- 10

10−10 100 (a) 10−11

10−12 ) ¯ ǫ k / ( ) h k 10−13 E Π(

A06 10−14 A10 Boussinessq B04 10−15 Shallow Water − 10−1 B08 ∼ k 2 10−16 100 101 102 100 101 102 k k 2.0 FIG. 9. (Color online) Power spectrum of h for simulations B04 and BSW 12. The former corresponds to a numerical (b) solution of the BQ model, while the later to a solution of the A10 ∼ −2 1.5 SW model. A k power law is indicated as a reference. ¯ ǫ value, or by nonlinear wave steepening in our case (which 1 0 . is ﬁnally regularized by the viscosity). Such a mechanism νZ/

2 is independent of the power injected by external forces. Of course, this can only hold in a region of parameter B08 0.5 A06 space, as in the presence of weak forcing and dispersion, B04 the solution in Eq. (25) is expected instead. In summary, based on the numerical results, the simu-

0 0 lations with weaker forcing and higher dispersion develop . 0 50 100 150 200 250 300 a spectrum compatible with the predictions from weak t turbulence theory, while the runs with stronger forcing or with less (or no) dispersion are compatible with dimen- FIG. 8. (Color online) (a) Energy flux (normalized by the sional analysis based on strong turbulence arguments. mean energy injection rate) as a function of k. For each simulation, a range of wavenumbers can be identified for which Π(k) remains approximately constant, and this range is in C. Comparison between SW and BQ models reasonably good agreement with the inertial ranges identified in Figs. 6 and 7. (b) Energy dissipation rate (normalized by the averaged in time energy injection rate) as a function of All simulations of the SW model belong to set B, as time for the same simulations. that is the set of runs that has negligible or no dispersion. All other sets have moderate dispersion, and as a result the flow dynamics cannot be captured by the SW model. tition) in the turbulent steady state. This implies that g Note that runs in set B are also the runs with an inertial is the only dimensional constant the spectra can depend range compatible with k−2 scaling. However, the BQ on. This is precisely how Phillips derived his spectrum. and SW runs in set B are∼ not identical. In this subsection With these assumptions in mind, it is easy to obtain we discuss the differences between these runs. the observed spectra. The energy spectrum has units of As an example of two runs with and without dispersive energy in the fluid column per unit surface per wavenum- effects, the power spectra of h for runs B04 and B 12 ber, E(k) h u2/k, and assuming E(k) gh k−α, from SW ∼ 0 ∼ 0 are shown in Fig. 9. Both simulations have the same dimensional analysis the only possible solution is parameters, except for the viscosity which is larger in −2 the simulation using the SW model. At small wavenum- E(k) gh0k . (38) ∼ bers, where dispersion is negligible, the spectra of the BQ The independence of the spectrum on the energy injec- and SW models coincides. For wavenumbers larger than tion rate suggests that the energy transfer between the 30, dispersion in the BQ model becomes important different scales must take place by a mechanism such as and≈ a bump (an accumulation of energy at small scales) wave breaking in the case of Phillips’ spectrum, which oc- develops. This accumulation in the BQ model results in curs when the slope of the surface is larger than a critical an increased dissipation (as dissipation is proportional 11

0.00208 mulation of energy in the spectrum near the dissipative B05 range is often termed “bottleneck”, and bottlenecks can 0.00206 BSW 12 have dissipative [55] or dispersive [49, 56] origins. In the former case, the accumulation results from the vis-

0.00204 cous damping of the triads at small scales, resulting in a decrease of the energy ﬂux. Such a viscous bottleneck should be visible also in the non-dispersive simulations, 0 00202 . and its absence in those runs indicates a dispersive ori-

h gin. In the latter case, the bottleneck arises from the 0.00200 increasingly harder to satisfy resonant condition for the wave frequencies, as the waves become faster at smaller 0.00198 scales. Models with a field filtered by the Helmholtz operator (as is the case for the BQ model, see Eq. 15) tend to develop a bottleneck (see [49] for a detailed description 0.00196 of its origin). A qualitative way to explain the tendency towards a flatter spectrum in the BQ model can be ob- 0.00194 0 π π 3π 2π tained by assuming that dispersion is strong enough for 2 2 the dispersive term to be balanced with the buoyancy x and with the non-linear terms in the BQ equations (i.e., all terms are of the same order). Then the energy spec- FIG. 10. One dimensional cut of the height h in the turbulent tra can depend only on both g and h0, and a possible 2 steady state of runs B04 and BSW 12, at the same time. The solution is E(k) gh0. A detailed study of the origin of former run corresponds to a numerical solution of the BQ this bottleneck is∼ left for future work. model, while the latter to a solution of the SW model. While At this point it is worth pointing out that when Ds 1 the long length scales show the same behavior in both runs, and dispersion becomes too strong, the Boussinesq≈ ap- note the BQ model has larger fluctuations at short length scales. Both runs were computed with a linear resolution proximation breaks down as more terms in the Taylor of N = 1024 grid points, and the fast fluctuations are well expansion in Eq. (11) should be preserved. As a result, resolved. the Boussinesq approximation is useful as long as Ds < 1 at the smallest excited scales in the system. On the other hand, from Fig. 3, if Ds . 0.15 the behavior of the sys- to k2E(k)), thus allowing us to simulate the system with tem in the inertial range is that of a shallow water flow smaller viscosity. This difference at large wavenumbers for all Reynolds numbers studied. is the most distinct feature in the two spectra in Fig. 9. As a result of the extra power at larger wavenumbers, dispersion in the BQ model results in more prominent D. Time-resolved spectra and non-linear dispersion small scale features, and in rapidly varying waves. As an relations example, Fig. 10 shows a transversal cut in the elevation field for runs B04 and BSW 12. The cuts are taken at Wavenumber spectra, as the spectra discussed so far, the same place and at the same time in both runs. Even give information of how energy is distributed in spatial though both simulations have the same behavior at large scales, but do not provide a quantitative estimate of how scales, at short length scales the BQ model presents fast much energy in the system is associated with wave mo- fluctuations. These fluctuations are well resolved (the cut tions. A frequency spectrum E(ω) is often obtained from corresponds to 1024 grid points), and there is no indica- the wavenumber spectrum E(k) using the dispersion re- tion that resolution is insufficient to resolve the sharp gra- lation (or vice versa). However, in systems that can sus- dients. In the BQ model, while the large scales may cor- tain both wave and vortical motions there is no clear respond to a shallow flow, as long as there is enough scale justification to use the dispersion relation to go from one separation, there will always be a wavenumber where the spectrum to the other. finite depth effects can be seen. Thus, the Boussinesq A quantification of the amount of energy in waves, and equations provide an interesting model to study weakly on whether non-linear effects change the dispersion rela- dispersive waves. tion of the system from the linear one, can be directly Regarding the accumulation of energy that leads to a obtained from the frequency and wavenumber spectrum flatter spectrum for high wavenumbers in some of the BQ E(k,ω) without any assumption. The spectrum E(k,ω) simulations (for several runs in set B as can be seen in can be computed by storing the Fourier coefficients of Fig. 7, but specially in the runs in set C), such an ac- the height hˆ(k,t) as a function of time (as well as the cumulation has been observed before in turbulent flows. Fourier coefficients of the velocity field), then computing As mentioned above, we verified that this accumulation the Fourier transform in time, and finally computing the is not the result of insufficient resolution (e.g., by com- isotropic power spectrum by averaging in the (kx, ky)- paring the runs with different grid points N). The accu- plane. To this end, several large-scale wave periods and 12

600 600

500 (a) 500 (a)

400 400

ω 300 ω 300

200 200

100 100

0 0 50 100 150 200 250 300 50 100 150 200 250 300 k k

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 600 600

500 (b) 500 (b)

400 400

ω 300 ω 300

200 200

100 100

0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 k k

10−15 10−12 10−9 10−6 10−3 100 103 106 109 1012 10−15 10−12 10−9 10−6 10−3 100 103 106 109 1012

FIG. 11. Power spectrum Eh(k,ω) for simulation B04. The FIG. 12. Power spectrum Eh(k,ω) for simulation A02. The darker regions correspond to larger power density, while the darker regions correspond to larger power density, while the lighter regions correspond to smaller power density. (a) Nor- lighter regions correspond to smaller power density. (a) Nor- malized power spectrum Eh(k,ω)/Eh(k). (b) Non-normalized malized power spectrum. (b) Non-normalized power spec- power spectrum. The white dashed line appearing in the trum. The white dashed line appearing in the bottom panel bottom panel indicates the linear dispersion relation from indicates the (non-dispersive) linear dispersion relation from Eq. (14). Note that as in this run dispersion is negligible, Eq. (4), and the white dash-dotted line indicates the BQ dis- the dispersion relation is almost that given by Eq. (4), and persion relation from Eq. (14). non-dispersive.

B04 that has a spectrum compatible with strong turbu- turnover times must be stored (to resolve the slowest fre- lence phenomenological arguments. There is also a tur- quencies in the system), with sufficient time resolution bulent broadening of the dispersion relation, also visible ∆t to resolve the fastest frequencies. In the analysis we in cross sections of the spectrum at different wavenum- show below, time series spanning at least three periods bers in Fig. 13. From this broadening, the characteristic of the slowest waves were used, and with time resolution time of non-linear wave interactions can be obtained, as − ∆t 3 10 4. was done in [57]. ≈ × Figures 11 and 12 show the power spectrum of the flow Some of the most important results in this paper are height Eh(k,ω) for simulations B04 and A02 respectively. associated with these two figures. First, note that in run The linear dispersion relations for shallow water flows B04 most of the energy is concentrated near a dispersion (Eq. 4) and for Boussinesq flows (Eq. 14) are also shown relation that, as dispersion is negligible, corresponds in as references, using the parameters from each run. Note practice to the non-dispersive shallow-water case (Eq. 4). both runs present an energy accumulation near the dis- All runs in set B have the same spectral behavior in −2 persion relation. This indicates most of the energy is in Eh(k,ω), and confirm that the k spectrum is ob- the waves, and remains there as time evolves. As we are served when dispersion is negligible∼ or absent (i.e., when not solving the equations for a potential flow, and the the flow is sufficiently shallow). Second, note that the system can develop vortical motions, this tells us that spectrum Eh(k,ω) in run A02 presents clear signs of dis- the non-linear energy transfer is mostly done between persive effects (i.e., most of the energy for large enough waves, and that the energy injected at large scales in k is concentrated over a curve that deviates from a linear wave motions is mostly transferred towards wave motions relation between k and ω), and this run displays a scaling at smaller scales and faster frequencies. This is needed in Eh(k) compatible with the weak turbulence prediction for weak turbulence to hold, but is also observed in run k−4/3. This behavior was observed in the other runs ∼ 13

103 satisfy the following dispersion relation, ∗ 102 k = 100 ∗ c0k 1 k = 200 Ω (Nk)= , (40) 10 N 2 2 ∗ h0k 0 k = 300 1+ 2 10 q 3N −1 10 which veriﬁes Eq. (39). However, the second branch in 10−2 Fig. 12 cannot be described by this dispersion relation 10−3 for any value of N up to 4, and thus they are not bound

−4 waves in the sense often used in oceanography.

Amplitude 10 Another explanation for the existence of these two −5 10 branches can be given by keeping in mind that at inter- 10−6 mediate wavenumbers slight variations in the ﬂuid depth 10−7 may trigger a transition in the waves from dispersive to

−8 non-dispersive (as the level of dispersion depends on the 10 0 100 200 300 400 500 600 700 ω product of the wavenumber with the surface height). In- deed, in the turbulent flow there are waves with short wavelengths which ride over long ones, that have a larger FIG. 13. Cross sections of Eh(k,ω) at different (and fixed) values of k = k∗ for run A02. Note the peaks and surrounding amplitude. For sufficient scale separation, the fast waves wavenumbers have most of the power. Note also the two see an effective depth that can be larger or smaller than peaks for k = 200, one corresponding to the shallow-water h0 depending on whether the wave is on a crest or a val- dispersion relation, and the other to the Boussinesq dispersion ley of the slow wave, generating in one case dispersive relation. waves, and in the other non-dispersive waves. We can estimate the variation in the effective dispersion at a given wavenumber k. In simulation A02, −3 in set A. h0 = 4 10 and the longer waves have an amplitude × −5 However, E (k,ω) for runs in set A presents yet an- δ 4 10 (as can be estimated, e.g., from the maxi- h ≈ × other interesting feature. As expected, for small k the mum value of the power spectrum of h). From the system dispersion is negligible and the energy is concentrated dispersion relation, over a straight line in (k,ω) space. At large k, as already mentioned, the effective dispersion relation is com- 2 2 2 1 2 2 ω = c0k 1 h0k , (41) patible with that of the linearized Boussinesq equations. − 3 But at intermediate wavenumbers two branches of the dispersion is controlled by the amplitude of the h2k2/3 dispersion relation can be observed, one that is compat- 0 term. Assuming that fast waves experience an effective ible with non-dispersive waves and another compatible depth h δ (where the sign depends on whether they with dispersive waves. When both branches are present, 0 are on a valley± or a crest), the variation in the dispersion their amplitudes are of the same order, as can be seen in is proportional to the difference between (h δ)2 and Fig. 13. 0 (h + δ)2. So, for this simulation, the variation− is around At first sight, the existence of these two branches could 0 4%, and when multiplied by k2, it is sufficient to explain be attributed to bound waves. Bound waves are small the two branches in E (k,ω) for k between 150 and amplitude waves which are bounded to a parent wave of h 250. ≈ larger amplitude. The waves are bounded in the sense that they follow the parent wave, i.e., they travel with the same phase velocity as the parent, and thus they fol- E. Time frequency energy spectra low an anomalous dispersion relation (see, e.g., a discussion of bound waves in the context of gravito-capillary waves in [58, 59]). The condition that they have the From the spectra in Figs. 11 and 12 the frequency spec- same phase velocity as the parent wave implies that they trum Eh(ω) can be easily obtained, simply by summing must follow a modified dispersion relation which verifies over all wavenumbers, Ω(k) = ω(k )k/k , where k is the wavenumber of the 0 0 0 E (ω)= E (k,ω). (42) parent wave. Bound waves result in multiple branches h h Xk in the E(k,ω) spectrum (and in multiple peaks in the frequency spectrum). Indeed, it is easy to show that for As already mentioned, in experiments and simulations k = Nk0, N =2, 3, 4,... , these multiple branches satisfy Eh(ω) is sometimes estimated instead from Eh(k) by using the dispersion relation in the form k = k(ω). Figure ΩN (Nk0)= Nω(k0) (39) 14 shows the power spectrum of h as a function of ω for simulations A02 and B04. In both cases, the spectrum (see, e.g., the discussion in [59]). Extending the analysis was calculated explicitly using Eq. (42), and also esti- in [59] to our case, bound waves in the BQ model should mated using the dispersion relation. For each run, the 14 two spectra show a very good agreement, which can be 100 expected as most of the energy is in the waves. The behavior of the inertial range in each run is also in good 10−1 agreement with the one found previously for Eh(k) in Sec. IV B.

10−2 10−9

10−10 (a) 10−3

10−11 Probability Density 10−4 Simulation

) Tayfun

ω 10−12 ( Skew Normal h 10−5 E −4 −3 −2 −1 0 1 2 3 4 5 10−13 h/σ PkEh(k, ω) −14 10 Eh(k(ω)) FIG. 15. (Color online) Probability density function of the −4 3 ∼ ω / values of h in simulation A06 (solid blue line). The dash- 10−15 dotted (red) line indicates a maximum likelihood ﬁt using 100 101 102 a skewed normal distribution, while the dashed (green) line ω corresponds to a maximum likelihood ﬁt for the Tayfun dis- 10−9 tribution.

10−10 (b) measuring large values of h than of small values. The shape can be adjusted by two distributions: We consider 10−11 a skewed normal distribution [60],

) 2 x ξ x ξ

ω 10−12 f(x)= φ − Φ α − , (43) (

h κ κ κ E 10−13 where κ is the so-called scale parameter (associated with the variance of the distribution), ξ is the location pa- P Eh(k, ω) k rameter (associated with the mean value), α is the shape 10−14 Eh(k(ω)) parameter (associated with the skewness), and −2 ∼ ω 10−15 1 − 2 100 101 102 φ(x)= e x /2, (44) √ ω 2π x 1 x Φ(x)= φ(t) t. = 1+erf . (45) FIG. 14. (Color online) Power spectrum of h as a function Z−∞ 2 √2 of the frequency for simulations (a) A02 and (b) B04. In We also consider a Tayfun distribution both cases, the spectrum was calculated by summing over all wavenumbers in the time and space resolved spectrum, ∞ − 2 − 2 2 e [y +(1 c) ]/(2s ) P E (k,ω), and also by using the dispersion relation given k h p(x)= y. , (46) by Eq. (14) to estimate the frequency spectrum from the Z0 πsc wavenumber spectrum Eh(k). As a reference, power laws − − ∼ ω 4/3 and ∼ ω 2 are shown in each case. The behavior is with c = 1+2sx + y2 and where s is the mean steep- in good agreement with the one found for Eh(k). ness of thep waves[61]. For run A06, and from a Maximum Likelihood Esti- mation method for the skewed normal distribution, the location parameter is ξ 1.00, the scale parameter is F. Probability density functions k 1.43, and the shape≈− parameter is α 1.94. For the≈ same run, and for the Tayfun distribution,≈ the mean We calculated the probability density function (PDF) steepness of the waves is s 0.15. This latter value is of the free surface height for diﬀerent simulations. Figure more relevant as the Tayfun≈ distribution is often used in 15 shows the PDF of h/σ for run A06, where σ is the oceanography and in experiments of surface waves. In standard deviation of the surface height. The probability this context, it is interesting to point out that experi- distribution is asymmetric, with a larger probability of ments in [37] found similar values for s. 15

This behavior (a PDF of h described correctly by both in all the simulations. In runs in set B, most of the energy a skewed normal distribution and a Tayfun distribution is concentrated in the vicinity the linear dispersion rela- with asymmetry to the left) was observed in all simulation for shallow water waves, which are non-dispersive. tions, no matter what set they belonged to. In runs in set A, the resulting non-linear dispersion relation obtained from Eh(k,ω) has two branches: one that corresponds to non-dispersive waves, and another corre- V. CONCLUSIONS sponding to dispersive waves. The two branches can be explained as the result of the superposition of rapidly We studied wave turbulence in shallow water flows varying waves which ride over slowly varying waves, the in numerical simulations using the shallow water and latter with sufficient amplitude to change whether the Boussinesq models. The equations were solved using former see a shallower or deeper fluid. grids up to 20482 points, and the parameters were varied (e) Independently of the differences between the runs, to study different regimes, including regimes with larger the probability distribution functions of h for the runs in and smaller Reynolds number, and larger and smaller dis- all sets is asymmetric, with larger probabilities of finding persion, while keeping the Froude number approximately larger values of h than smaller values. The probabil- the same. We summarize below the main conclusions fol- ity distribution functions can be approximated by both lowing the same ordering as in the introduction: a skewed normal distribution and a Tayfun distribution (a) As in previous experimental and observational [61]. In the latter case, the only parameter of the dis- studies [35, 36], we found that the flows can be clas- tribution, the mean steepness of the waves, has values sified in different sets depending on the value of the compatible with those found in observations and experi- Reynolds number (i.e., on the strength of the nonlinear- mental studies (see [37]). The obtained probability den- ities) and on the level of dispersion (associated with the sity functions also indicate limitations in the hypothesis fluid depth). A first set (A) has smaller Reynolds num- of Gaussianity of the fields assumed in early theories of bers and stronger dispersion, a second set (B) has larger weak turbulence. However, extensions of the theory to al- Reynolds numbers and weaker or negligible dispersion, low for non-Gaussian distributions exist and can be found and a third set of runs seems to be transitional between for example in [62] and [63, 64]. the two. All the results presented here were obtained solving (b) Runs in sets A and B have different power spec- numerically equations that do not assume that the flow tra of the surface height. Runs in set A, with stronger is inviscid or irrotational, and with realistic terms for dispersion, present a spectrum compatible (within sta- the viscous dissipation. We believe this approach can be −4/3 tistical uncertainties) with Eh(k) k . This is the useful to compare with experiments, as in experiments spectrum predicted by weak turbulence∼ theory for the vorticity can develop in the flow, and viscosity cannot be Boussinesq equations [34]. Runs in set B with negligible neglected. or zero dispersion (i.e., for a shallower flow) show a spec- −2 trum compatible within error bars with Eh(k) k . This spectrum can be obtained from phenomenological∼ ACKNOWLEDGMENTS arguments coming from strong turbulence [27]. The runs in set C have no discernible inertial range. The authors would like to thank Prof. Oliver Buhler (c) The Boussinesq (dispersive) model tends to develop and the anonymous referees for their useful comments. more power in waves with short wavelengths than the The authors acknowledge support from grants No. PIP shallow water model. This is associated with the de- 11220090100825, UBACYT 20020110200359, and PICT velopment of a bottleneck for large wavenumbers in the 2011-1529 and 2011-1626. PDM and PJC acknowledge energy spectrum. support from the Carrera del Investigador Científico of (d) Inspection of the wave and frequency spectrum CONICET, and PCdL acknowledges support from CON- Eh(k,ω) confirms that most of the energy is in the waves ICET.

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